Solve for v |2+7v|+11=27

$| 2 + 7 v | + 11 = 27$

Step 1

Move all terms not containing $| 2 + 7 v |$ to the right side of the equation.

Subtract $11$ from both sides of the equation.

$| 2 + 7 v | = 27 - 11$

Subtract $11$ from $27$ .

$| 2 + 7 v | = 16$

Step 2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$2 + 7 v = \pm 16$

Step 3

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$2 + 7 v = 16$

Move all terms not containing $v$ to the right side of the equation.

Subtract $2$ from both sides of the equation.

$7 v = 16 - 2$

Subtract $2$ from $16$ .

$7 v = 14$

Divide each term in $7 v = 14$ by $7$ and simplify.

Divide each term in $7 v = 14$ by $7$ .

$\frac{7 v}{7} = \frac{14}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 v}{7} = \frac{14}{7}$

Divide $v$ by $1$ .

$v = \frac{14}{7}$

Simplify the right side.

Divide $14$ by $7$ .

$v = 2$

Next, use the negative value of the $\pm$ to find the second solution.

$2 + 7 v = - 16$

Move all terms not containing $v$ to the right side of the equation.

Subtract $2$ from both sides of the equation.

$7 v = - 16 - 2$

Subtract $2$ from $- 16$ .

$7 v = - 18$

Divide each term in $7 v = - 18$ by $7$ and simplify.

Divide each term in $7 v = - 18$ by $7$ .

$\frac{7 v}{7} = \frac{- 18}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 v}{7} = \frac{- 18}{7}$

Divide $v$ by $1$ .

$v = \frac{- 18}{7}$

Simplify the right side.

Move the negative in front of the fraction.

$v = - \frac{18}{7}$

The complete solution is the result of both the positive and negative portions of the solution.

$v = 2, - \frac{18}{7}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$v = 2, - \frac{18}{7}$

Decimal Form:

$v = 2, - 2.\overline{571428}$

Mixed Number Form:

$v = 2, - 2 \frac{4}{7}$

Solve for v |2+7v|+11=27

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